Platonic Solids

1
Platonic Solids

• Nader Abbasi

2
Introduction

• Since the earliest times, human mind has been
  fascinated by the five objects known as Platonic
  Solids. These are convex solids which have
  congruent faces and are called regular polyhedra.
  Due to symmetry and aesthetic beauty, these
  objects have been subject of study of the best
  human minds for thousands of years.

3
Platonic Solids

• Platonic solids is another name for regular
  polyhedra.
• Regular polyhedra have regular polygon faces and
  identical vertices.
• The five Platonic solids are the only convex
  regular polyhedra.

4
Euclids proof

• At least three polygons are needed to make a
  solid angle.
• The smallest regualr polygon is an equilateral
  triangle.
• Such an angle can be constructed with 3,4,and 5
  equilateral triangles.
• With 6 equilateral triangle the result lies flat.

5
Square Solids

• The next regular polygon is square
• 3 squares around a point forms a solid angle, but
  with 4 squares the result is a flat surface. 3
  squares around a point is the only possible case
  for making a solid angle with squares.

6
Pentagonal Solid

• The next polygon is the pentagon, 3 of which
  around a point make a solid angle.
• There is no room for 4 pentagons, even to lie
  flat. Like the square, 3 is the only combination
  to form a solid angle.

7
Polyhedra Limit!

• The number of regular polyhedra is limited to 5.
• The next regular polygon is the hexagon. 3
  hexagon lie flat and do not form a solid angle
  and there is no other regular polygon that 3 of
  them can meet at a point and form a solid angle.